Let $\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s),\qquad$ $F(x)=\frac{\xi'}{\xi}\!\left(\frac1{1-x}\right)=\sum_{m\ge0}f_mx^m,$ and define the symmetric Toeplitz--Hankel coefficients $g_{ij}=f_{|i-j|}-f_{i+j+1}+\delta_{ij}f_0.$ The adjacent matrices $M_n=\begin{pmatrix}g_{nn}&g_{n,n+1}\\g_{n,n+1}&g_{n+1,n+1}\end{pmatrix}$ form the local family in a previously established criterion equivalent to the Riemann hypothesis. We prove an unconditional finite-range positivity theorem for this family. The zero-pair moment representation of $g_{ij}$ expresses $M_n$ as a sum of rank-one polynomial atoms. A finite verification of the Riemann hypothesis up to height $H$ then splits this sum into a positive-semidefinite verified part and an unrestricted high-zero tail. Two rigorously isolated low zeros provide a positive core; every zero above $H$ is controlled in operator norm by an explicit zero-counting estimate. For the polynomial recurrence $B_0(t)=1,\quad B_1(t)=3-t,\quad B_{m+1}(t)=(2-t)B_m(t)-B_{m-1}(t),$ we derive the exact oscillatory form $B_m(4\sin^2\alpha)=\frac{\sin((2m+1)\alpha)}{\sin\alpha}$ and a closed formula for its two-point Christoffel--Darboux kernel. Using the Platt--Trudgian verification height $H=3{,}000{,}175{,}332{,}800$, the first two LMFDB/Platt zero intervals, and directed-rounding MPFR arithmetic, we certify the kernel away from zero at every integer level $0\le n\le99{,}999$. The resulting positive-core lower bound exceeds the adversarial high-zero tail bound by more than ten orders of magnitude. Consequently $M_n\succ0\qquad(0\le n\le99{,}999),$ or equivalently the first $100{,}000$ adjacent Toeplitz--Hankel determinants are strictly positive. No assumption is made about zeros above the verified height. To the best of our knowledge, this is the first finite-height transfer theorem for this adjacent Toeplitz--Hankel family and the first rigorous certification of its initial $100{,}000$ strict inequalities. This is not a proof of the Riemann hypothesis.
We apply the Omuo Genesis Engine, a geometric knowledge synthesis platform operating on the E8 lattice, to map the structural landscape of known approaches to the Riemann Hypothesis. Approximately 250 concepts spanning analytic number theory, spectral theory, algebraic geometry, quantum chaos, p-adic analysis, and the Langlands program were encoded as complex phasor vectors in C^1024 and iteratively bound through five ouroboros (self-feeding) cycles. The resulting manifold (2,379 nodes, 199 bridges, 113 unique E8 axes) identifies the Selberg Trace Formula as the central nexus of the RH landscape, appearing nine times from independent parent combinations. The terminal structure is a fixed-point cycle between the Selberg Trace Formula, the Spectral Determinant, and the Semiclassical Quantization Condition. The engine's deepest bridge proposes deformation invariance of the spectral determinant as the key mechanism: the zeros lie on the critical line because they cannot be moved without breaking a topological invariant. Novel structural connections include bridges between Arakelov heights and spectral determinants, between braid monodromy and trace identities, and between spectral deformation and Selmer groups. These are presented as structural observations from geometric synthesis, not as mathematical proofs.
The first machine-checked formalization, in any proof assistant, of any component of the Birch-Swinnerton-Dyer (BSD) conjecture pipeline. No prior Lean, Coq, or Isabelle project has formalized Silverman height bounds, Gross-Zagier-Kolyvagin data structures, or the logical architecture of BSD generator search. No new number theory is proved. The algorithms formalized are the engines inside Cremona's mwrank and SageMath. The contribution is the formalization itself and the foundational analysis it enables. Three results that did not previously exist in formal mathematics: (1) A machine-checked axiom/theorem boundary for BSD. The formalization identifies exactly which ingredients must be axiomatized (Gross-Zagier, Kolyvagin, Silverman bound, positive-definiteness) and which can be proved constructively (height bound chain, finite grid membership, search space finiteness). This is the blueprint for any future formally verified BSD solver: the deep analytic theorems interface with type theory through a single chokepoint (the real-valued Silverman bound), and everything below that chokepoint is verified. (2) A logical characterization of the Archimedean/p-adic dichotomy. The positive-definite Archimedean metric (u = 1) is identified as the exact logical modality lowering search complexity from Pi^0_1 (unbounded, MP) to Delta_0 (bounded verification, BISH). This foundational statement does not appear in the classical literature -- Cremona and Watkins use height bounds as engineering, not as a theorem in reverse mathematics. (3) A logical explanation of the exceptional zero pathology. The p-adic BSD exceptional zero (Mazur-Tate-Teitelbaum) is usually explained analytically: trivial zeros of p-adic L-functions, extra Euler factors. This formalization gives a logical explanation: the p-adic canonical height is not positive-definite, so the MP-to-BISH conversion fails. The search remains unbounded because the metric lacks the topological property needed for logical reduction. This re-reading of a classical analytic obstruction as a failure of logical reducibility is, to our knowledge, new. The axiom budget is minimal -- removing any one ingredient breaks the proof chain -- characterizing the necessary logical interface between analytic number theory and formal verification. This is the first application of constructive reverse mathematics to a Clay Millennium Problem. Formalized in Lean 4 + Mathlib with zero sorry's and zero custom axiom declarations. All analytic axioms enter as Prop-valued hypotheses in a BSDRankOneData structure. Axiom audit: every theorem depends only on [propext, Classical.choice, Quot.sound] (standard Mathlib infrastructure for the reals). Package contains compiled PDF (10 pages), LaTeX source, and complete Lean 4 source (7 files, ~725 lines) buildable with lake build.
We investigate the mean number of real zeros over an interval $[a,b]$ of a random trigonometric polynomial of the form $\sum_{k=1}^n a_k \cos(kt)+b_k \sin(kt)$ where the coefficients are i.i.d. random variables. Under mild assumptions on the law of the entries, we prove that this mean number is asymptotically equivalent to $\frac{n(b-a)}{Ï€\sqrt{3}}$ as $n$ goes to infinity, as in the known case of standard Gaussian coefficients. Our principal requirement is a new Cramer type condition on the characteristic function of the entries which does not only hold for all continuous distributions but also for discrete ones in a generic sense. To our knowledge, this constitutes the first universality result concerning the mean number of zeros of random trigonometric polynomials. Besides, this is also the first time that one makes use of the celebrated Kac-Rice formula not only for continuous random variables as it was the case so far, but also for discrete ones. Beyond the proof of a non asymptotic version of Kac-Rice formula, our strategy consists in using suitable small ball estimates and Edgeworth expansions for the Kolmogorov metric under our new weak Cramer condition, which both constitute important byproducts of our approach.
Yiannis Sakellaridis and Akshay Venkathesh have determined, when the group\n$G$ is split and the field $\\F$ is of characteristic zero, the Plancherel\nformula for any spherical space $X$ for $G$ modulo the knowledge of the\ndiscrete spectrum.\n The starting point is the determination of good neighborhoods at infinity of\n$X/J$, where $J$ is a small compact open subgroup of $G$. These neighborhoods\nare related to "boundary degenerations" of $X$. The proof of their existence is\nmade by using wonderful compactifications.\n In this article we will show the existence of such neighborhoods assuming\nthat $\\F$ is of characteristic different from 2 and $X$ is symmetric. In\nparticular, one does not assume that $G$ is split. Our main tools are the\nCartan decomposition of Benoist and Oh, our previous definition of the constant\nterm and asymptotic properties of Eisenstein integrals due to Nathalie Lagier .\n Once the existence of these neighborhoods at infinity of $X$ is established,\nthe analog of the work of Sakellaridis and Venkatesh is straightforward and\nleads to the Plancherel formula for $X$.\n
<!-- *** Custom HTML *** --> We associate to a regular system of weights a weighted projective line over an algebraically closed field of characteristic zero in two different ways. One is defined as a quotient stack via a hypersurface singularity for a regular system of weights and the other is defined via the signature of the same regular system of weights. The main result in this paper is that if a regular system of weights is of dual type then these two weighted projective lines have equivalent abelian categories of coherent sheaves. As a corollary, we can show that the triangulated categories of the graded singularity associated to a regular system of weights has a full exceptional collection, which is expected from homological mirror symmetries. The main theorem of this paper will be generalized to more general one, to the case when a regular system of weights is of genus zero, which will be given in [5]. Since we need more detailed study of regular systems of weights and some knowledge of algebraic geometry of Deligne–Mumford stacks there, the author write a part of the result in this paper to which another simple proof based on the idea by Geigle–Lenzing [2] can be applied.
In this note we point out and fill a gap in the proof by Julg-Kasparov of the Baum-Connes conjecture with coefficients for discrete subgroups of $\op{SU}(n,1)$. The issue at stake is the proof that the complex powers of the contact Laplacian are element of the Heisenberg calculus. In particular, we explain why we cannot implement into the setting of the Heisenberg calculus the classical Seeley's approach to complex powers.
1. A. Weil [3] constructed a universal distribution t on the Weil group.The values of I at various test functions give the contributions from the zeros of L-functions which appear in the .explicitformulas.In this note, we shall construct a universal distribution zi on GL(n) and prove the explicit formula for automorphic L-functions using z/ when n-2.For n 2, to derive such a result, we must assume certain property of characters of infinite dimensional representations of GL(n) over a local field.This property, formulated as Conjecture, seems to lie slightly beyond our present knowledge of harmonic analysis.The distributions A have striking formal resemblance to Weil's one.Furthermore they are related to each other so that zl is the "direct image" of z/ for m n.This is a pleasant fact since we think that a discovery of new functorial properties related to zeros of zeta functions would be crucial for the proof of the Riemann hypothesis.
This thesis is an attempt to generalise to the odd orthogonal group Γ_K, over an Infinite field K not of characteristic two, the work of Schur [S], and more recently Green [G], on the general linear group G_K using the approach of Weyl [W] in characteristic zero. The special feature here is that we treat Γ_K as merely a group of matrices defined by the vanishing of polynomials in its coefficients (the classical view) rather than a group generated by elements derived from an associated Lie algebra, the approach used initially by Chevalley and adopted by most authors in recent times. \n \nAfter generalising Green's [G] Schur algebra for G_K to Γ_K in §0 we prove in §1 Chevalley's famous theorem on the 'Big Cell' in G_K and then, by an easy extension, prove it for the Big Cell in Γ_K. Chevalley's original proof uses representations of Lie algebras, ours requires nothing but a little knowledge of the coordinate ring K_+[G] of all 'polynomial' functions on G_K . We define K[Γ], the coordinate ring of Γ_K, to be the space of all polynomial functions on G_K restricted to Γ_K and in §2 give a generating set of the kernel of the restriction map ψ_K:K_+ [G]→K[Γ]. This generalises Weyl's result in characteristic zero. In §3 we use this result to show that the family, or 'scheme', of rings K[Γ] (K varying over all infinite fields not of characteristic two) is 'defined over Z' ; in fact K[Γ] is naturally isomorphic to K θ Z[Γ_Q], where Z[Γ_Q] is the subring of Q[Γ] spanned by 'monomial' functions. This enables us to formulate a 'modular' representation theory for Γ which connects polynomial representations of Γ_Q with those of Γ_K. \n \nIn §4 we investigate the Schur algebras of Γ_Q following Weyl [W] and in §5 find a complete set of irreducibles for each of them, once again following the lead of Weyl. In §6 we attempt to 'reduce' these modules modulo p to obtain 'Weyl' modules for Γ_K, a task only partially completed.
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